Can the projection of **w** onto **v** point in the *opposite*
direction of **v** ?

Yes, as it did in that example.

The diagram shows:

w= (4, 2.5, 2.5)^{T}— red vector

and

v= (5, 0, 3.1)^{T}— green vector.

Mentally project **w** onto **v**.
(Hint: remember that the length of **v** does not matter; only its orientation.
So mentally "drop" the arrowhead of **w** onto the dotted line).